Pawel TopaAGH University of Science and Technology

Poland

1. Introduction

Most historical sources say that Cellular Automata were discovered by John von Neumannand Stanislaw Ulam in the forties of the twentieth century. Von Neumann was investigatingthe problem of self-reproducing systems in biology. His initial kinematic model appeared to betoo complicated and he followed the suggestion from Stanislaw Ulam, who at the same timeworked on problems of crystal growth. Von Neumann and Ulam defined an abstract universein the form of two-dimensional regular mesh with interacting entities Neumann (1966).In the seventies mathematician John Conway published his set of rules called "Game ofLife". His work was popularized by Martin Gardner in the pages of Scientific American. Therules of "Game of Life" implemented on computers showed an amazing world of living anddying automata. Its dynamics appeared to be so surprisingly complex that it became areaof intensive studies. Vast amount of work concentrated on searching for initial conditionsthat evolve into non-trivial behaviour of Cellular Automata. One of the most importantachievement was proving that "Game of Life" can simulate Universal Turing Machine i.e. canact as an abstract model of any computer.In the eighties Stephen Wolfram spent part of his scientific career on investigating propertiesof one-dimensional Cellular Automata. His important achievement was definition of 256 rulesfor one-dimensional Cellular Automata. He also introduced first classification scheme basedon long period observation. The crowning achievement of his works was a book "A New kindof Science" Wolfram (2002). In this monumental work Wolfram outlines a new paradigm formodelling complex phenomena with Cellular Automata approach.Scientific research related to Cellular Automata were not only limited to abstract mathematical“toy”. Toffoli, Margoulus and Fredkin postulated application of cellular automata formodelling physical systems Chopard & Droz (1998); Toffoli & Margolus (1987). They noticedthat physical laws can be encoded in rules of interaction in temporally and spatially discreteuniverse.The milestone in Cellular Automata theory was the lattice gas model HPP developed byHardy et al. (1976). The model, which is in fact the Cellular Automata, consists of a simpleand fully discrete system of moving and colliding particles. Although, the idea of latticegas automata was invented independently, the Cellular Automata theory provided clearconceptual framework. HPP model and his successor FHP Frisch et al. (1986) was successfulin modelling complex phenomena as flows in porous media, spreading of a liquid droplet andwetting phenomena.It is difficult now to figure out all the areas where the Cellular Automata paradigm is applied:

Network Systems Modelled by Complex Cellular Automata Paradigm

12

www.intechopen.com

• biological processes: tissue growth Wang et al. (2008), angiogenesis Markus et al. (1999),cancer development Reis et al. (2009) etc.

• chemical processes Chopard & Droz (1998); Kier (2000); Kier et al. (1998),

• granular flow Baxter & Behringer (1991); Masselot & Chopard (1996),

• pedestrian dynamics and traffic flow Blue et al. (1997); Was (2005),

• geological processes D’Ambrosio et al. (2001).

“Cellular Automata Modelling of Physical Systems” by Chopard & Droz (1998) containsvery profound review of various applications of Cellular Automata approach for modellingphysical systems.Cellular Automata in their classic meanings have rather limited application area. On theother hand their simplicity in computer implementation is irresistible temptation. A lot ofextensions were introduced to their definition. The set of states was extended by introducingfloating point numbers instead of logical values. The neighbourhood notion was extendedby introducing new schema (von Neumann, Moore and Margoulus neighbourhoods) andrange bigger then one cell. The homogeneity of Cellular Automata were also broken. Insome applications synchronous updating were also replaced by other asynchronous schemas.Mesh types other then rectangular were also applied in some models. Some extensions makethat model is so far from classic Cellular Automata definition that it could be classified as adifferent modelling tool. The Cellular Potts model Graner & Glazier (1992) and Agent-Basemodels Wooldridge (2009) can be treat as generalized Cellular Automata.Cellular automata paradigm fits perfectly to paradigm of computation on digital computers.When we use Cellular Automata approach we do not have to struggle with round-offerrors, truncation errors, numerical stability and lots of other problems that appear when weimplement methods of numerical analysis on computers. Rules of local interaction can beeasily and unambiguously defined by using programming languages constructions (if ... then... , while ... do ...). Algorithms and data structures necessary to develop Cellular Automatamodels are relatively simple and give great opportunity to efficient implementation.Moreover, Cellular Automata are inherently parallel. Cellular Automata definition states thatcells have to be processed simultaneously. Computer implementation have to simulate thisby using additional temporary data structure. In the 80s and 90s of 20th century a lot ofwork had been devoted to constructing hardware implementation of Cellular Automata.The most famous were Cellular Automata Machines built by Tommaso Toffoli and NormanMargoulus (CAM-6 i CAM-8) Margolus et al. (1986); Toffoli & Margolus (1987). There werealso successful attempts to implement CA on transputer machines Cannataro et al. (1995);Somers & Rem (1989) however this architecture did not succeed an fallen into oblivion. Finally,these interesting researches were abandoned in favour of flexibility of programming solutions.Recent research in GPGPU computing methods shows that Cellular Automata can run veryeffective on graphic processors Gobron et al. (2010).In this chapter I present applications of Cellular Automata paradigm for modellingdynamically evolving system with networks structure that function as transportationpathways. The following section presents phenomena that was studied and modelled by theauthor. Next I present the general idea of Graph of Cellular Automata that combine of CellularAutomata and graphs. In further section I present two models that employed Graph CellularAutomata:

• model of anastomosing river system,

• model of tumour induced angiogenesis.

260 Cellular Automata - Simplicity Behind Complexity

www.intechopen.com

I conclude the chapter with discussion on extensions and modifications that make CellularAutomata useful in modelling physical systems.

2. Network systems modelled with cellular automata

Cellular Automata fit best for modelling systems that are characterized by inner structuralhomogeneity and regularity, for example crowd of pedestrians or fluids. When we look forCellular Automata that are able to capture complex systems with interacting parts that have adifferent structure, we have to extended classic definition and combine it with other methods.One of the phenomena that we observe in everyday life are transportation systems locatedin consuming or producing environments. Transportation networks are transfer pathwaysfor various substances which are produced or absorbed by surrounding environments. Thesesubstances can also diffuse through the environment. We can enumerate two examples of suchthe systems that we meet in everyday life:

• Vascular systems — blood vessels transport nutrients and oxygen which penetrate into thetissue and nourish it. Products of metabolism penetrate from tissue to blood vessels andthey are removed from the body. Tissue can influence the vascular network by producingvarious substances called angiogenic factors that stimulate angiogenesis (process offormation of blood vessels).

• River systems — rivers transport organic and mineral materials from upper, erosionalpart of river system to lower part were the loads are deposited. In whole the basin, thesurrounding environment (shape of terrain) are modified by river (erosion and deposition).Changes in terrains influence river systems by changing the routes of channels as well asthe shape of network.

Such the systems are characterized by coexistence two parts having completely differentstructure. Environment has (in general) regular, uniform structure where substances aredistributed or collected in whole area or volume. Regular mesh with dense network ofconnection between cells can accurately approximate these processes. In the networks,processes are rather limited to the paths defined by their branches. Thus, we can easilysimplify their representation by using graph.Graph of Cellular Automata is a hybrid modelling method employing Cellular Automatatheory and some elements of graph theory. Cellular Automata is a basis that is used toconstruct graph. Some cells from mesh of automata are selected and they establish set of nodesin graph. An additional relations of neighbourhood are established between these cells, whatconstitute set of edges. Figure 1 illustrates this structure.

Fig. 1. Graph of Cellular Automata, from Topa & Dzwinel (2009)

261Network Systems Modelled by Complex Cellular Automata Paradigm

www.intechopen.com

Regular mesh and graph are processed separately and often in different manner (e.g.synchronously versus asynchronously). Cells that belong to the graph act as connectionsbetween these two systems and provide interaction between them. State of these cells aretransfer way for results of environment simulation on regular mesh of Cellular Automata tothe graph structure representing network system (see Fig. 2).

Fig. 2. Interaction between Cellular Automata and Graph constructed over CellularAutomata is transfered through the states of cells. States of all the cells in the mesh (Sij) caninfluence evolution of network system encoded in graph. Network system influencesenvironment through the cells that belong to graph (Sij ∈ VG)

The paradigm of Graph of Cellular Automata was applied for simulating two phenomena:

• Anastomosing river systems Topa & Dzwinel (2003); Topa et al. (2006); Topa & Paszkowski(2001)

• Tumour Induced Angiogenesis Topa (2006; 2008); Topa & Dzwinel (2009)

2.1 Rivers and anastomosing river systems

Anastomosing rivers are interesting dynamic system created as a result of interaction betweenconstructive and destructive processes. Anastomosing systems are formed in middle part ofriver system. Slope on these area is very small — approximately 10cm per 1km. Flow rateis also very small, river do not erode terrain. Organic and non-organic materials carried byriver are mostly transported through this part of river and deposited later in lower reaches.Waters rich of nutrients penetrate soil in surrounding banks. Fertile areas near river channelsare characterized by a significant increase in vegetation. Products of plant decay deposit onthis area as peat (approximately 1-1.5 mm/year Gradzinski et al. (2003)). Peat accumulationlasting for thousands years may produce a peat bog with a depth of several meters.The processes that occur on river channel banks are crucial for creation anastomosingsystem. Figure 3 illustrates these processes. It presents cross-section of river channel. Therate of peat accumulation depends on the distance from river channel. Water penetrates soilnearby the channel and supplies nutrients. Nutrients are gathered by plants that vegetateon the river banks. On the areas that are more distant from water, the vegetation are lessabundant than on the areas that directly neighbour to channel. Consequently, the rate of peatdeposition decreases with distance from channel. Deposition on river banks is accompaniedby deposition on river bottoms. As a result the whole river channel is raised (see in Fig 3). Thewater level is raised above the average valley ground level. Sometimes channel can be alsopartly blocked by clusters of floating plants, tree trunks etc. It is metastable state which canbe easily destroyed. High levels of water that usually occur during spring floods can causebreaking river banks above obstacles. River can create bypass channel that evade narrownesson primary channel. The shape of terrain in valley usually force new channel to join primarychannel somewhere downstream.The routes of new channels run through local depression areas. Evolution of new channelproceed in the same way as the parent channel. Old channels may also disappear gradually.The system evolves to dynamic metastable state in which whole the area of the valley is

262 Cellular Automata - Simplicity Behind Complexity

www.intechopen.com

Fig. 3. Crossection of anastomosing river channel

equally supplied by river systems. As a result the bottom of valley raises covered by thick(several meters) layer of peat-bog.Anastomosing rivers create very complex pattern of forking and joining channels. Fig. 4presents part of real network of river Narew located in eastern Poland (Central Europe). It isrelatively small anastomosing river comparing to huge anastomosing systems created by riverOb in Syberia (anastomising system can be observed from space) and Okawango in Africa.

2.2 Vascular system and tumour-induced angiogenesis

Angiogenesis is the process of formation of vascular network Carmeliet (2005).Physiologically this process occurs during embryogenesis. In mature organisms it oftenaccompanies to cancer development and this pathological process is called tumour-inducedangiogenesis. Inhibition of tumour induced angiogenesis is one of the treatments applied inoncology.Tumour growth is a multi-stage and multi-scale process that starts with the loss of control ofcell proliferation Carmeliet (2005). Initially, tumour is in avascular state and draws nutrientsfrom surrounding vessels. At this stage cluster of tumour cells can reach size aproximately 1- 3 mm. Tumour remains in steady state with balance between dying and proliferating cells.Without blood vessels, tumours cannot grow beyond a critical size and invade other regionsof body. Tumour induced angiogenesis starts when the production of pro-angiogenic factorsovercomes other forces that kept the angiogenesis quiescent so far.Oxygen and nutrients penetrate the tissue only in a certain distance from the vessel. Distantcells, influenced by metabolic stresses, synthesise several angiogenic stimulators includingVEGF (Vascular Endothelial Growth Factor) and PDGF-BB (Platelet-derived growth factor),Ang2 and NOS (Nitric Oxide Synthase) Carmeliet (2005); Steve et al. (2004). Stimulatorsmigrate towards the nearest blood vessels. When they reach vessel, the endothelial cells (ECs)that lines the wall of this vessel are activated. They start to proliferate and migrate towardsthe tumour cells attracted by VEGF and other stimulators. The wall of the parent blood vesselbecomes degraded and it opens to a new capillary. Migrating and proliferating ECs form ahollow tube-like cavity (the lumen), which are stabilised later by smooth muscle cells andperycites.There are many differences between normal and tumour induced angiogenesis. Temporal andspatial expression of angiogenic factors are not well coordinated, what follows to non-uniformvascular development. As a result, new vessels form a highly chaotic and disorganisednetwork Tonini et al. (2003). Tumour vessels are deprived various protective mechanisms i.e.perivascular cells that protect vessels from changes in oxygen or hormonal balance. Moreoverthe lumen does not have proper construction. Their walls may be partially constructed fromcancer cells. They have also pathological form, e.g. they are thin and permeable, their diameter

263Network Systems Modelled by Complex Cellular Automata Paradigm

www.intechopen.com

Fig. 4. Pattern of river channels in anastomosing part of Polish river Narew, from Topa et al.(2006)

changes abruptly. Proliferating cancer cells can also crash the vessels. All it makes that newlyformed capillaries are still unable to supply starving tumour cells with oxygen and nutrients.Inhibition of tumour-induced angiogenesis is one of the most promising strategy inanti–cancer therapy Carmeliet (2005); Ferrara & Kerbel (2005). Tumour cells deprived ofnutrients and oxygen undergo necrosis. Moreover, the lack of vascular system prevents cancercells from invading to other tissues (there is no routes, which tumour cells can use to transferto other part of body). However, clinical tests show that none of the tested inhibitors succeedin broad range types of cancers Carmeliet (2005). Monotherapies fail because angiogenesisis controlled by very complex balance of stimulators and inhibitors. Therefore, furtherinvestigations have to concentrate on studies including wider range of angiogenic factors.

3. The models

The Graph of Cellular Automata modelling tool was initially invented for modellinganastomosing river systems Topa & Paszkowski (2001). Pure Cellular Automata modelappeared too computationally demanding and it could not reflect global evolution of the

264 Cellular Automata - Simplicity Behind Complexity

www.intechopen.com

!"#

!!"# !"#$%&'(!')%*&%*+* ,'"*)% &*-.

,*/#""*)0%+. ." %1#&2%3"!!$

.4$!&2."#*"%,."" % /.)0,#&. %*4$% (!!&2%(' ,".%,.""

> 2$3##

%&#'&(")*+,'+-*,."/0.1'(2

3-.(-02-34-(#-05,6,17

%&#'&(")*+,'+-*,."/0.1'(2

(#-)*&#!4%*4$/)!"#5.)*&#!4%!5%.4$!&2."#*"%,.""

(*"5!)(.$%&'(!')%,*//#"*)

Fig. 5. Stages of tumour induced angiogenesis

system Topa et al. (2006). The model called MANGraCA (Model of Anastomosing Networkwith Graph of Cellular Automata) is constructed in the following way:

Cellular Automata is applied to model peat bog in the river valley and the followingprocesses are reflected in this part of the model:

• nutrient distribution,

• growth of peat layer.

Graph of Cellular Automata is applied to model network of river channels. The followingprocesses are implemented in this module:

• flow in river channels,

• decreasing of channel throughputs caused by peat bog development andsedimentation,

• branching, routing of new channels, anastomosing.

Cells that belong to the graph are the source of nutrients from where they are distributedover the regular mesh of automata that represent peat bog.

In more formal way the model can be defined as follows:

CAMANGraCA = 〈Z2, XK , GCA, S, δK , δG〉 (1)

• Z2 — a collection of cells ordered as a square or hexadecimal mesh of Z × Z cells,

• XK — neighbourhood for the (i, j) cell in the regular mesh of automata (Mooreneighbourhood is used in the model),

• GCA — a planar, directed and acyclic graph defined as (VG, EG), where VG ⊂ Z2 andEG ⊂ Z2 × Z2 are finite sets of vertices and edges, respectively,

265Network Systems Modelled by Complex Cellular Automata Paradigm

www.intechopen.com

• S — is the set of state vectors corresponding to each cell: S = Sm × Sg:

1. states for cells in regular mesh Sm = (g, n, p):

(a) g — initial altitude,

(b) p — thickness of peat layer,

(c) n — nutrients density,

2. states for graph cells Sg = ( f , r):

(a) f — flow rate,

(b) r — throughput;

• δK : (gt, nt, pt) → (gt+1, nt+1, pt+1) — set of rules applied synchronously for each cell inZ2.

• δG : ( f t, rt) → ( f t+1, rt+1) — set of rules applied for each cell that belong to graph.

Fig. 6. Hierarchical structures of anastomosing pattern generated with model ofanastomosing river.

The simulation program consists of two modules:

1. Cellular Automata Module — it calculates and update cells in regular mesh of cellularautomata. It simultaneously apply defined rules of local interactions (δK) to each cell in themesh and calculated new states.

• Nutrients density (nij) is calculated as average density in neighbourhood. Cells thatbelong to graph have maximum density.

• Thickness of peat layer (pij) is increased proportionally to the nutrients density.

2. Graph of Cellular Automata Module — it updates states of the cells that belong to thegraph.

• Throughput rij is decreased proportionally to thickness of peat bog layer (pij) inneighbourhood with random modifier.

• Flow rate fij is updated asynchronously starting from “root” node of the network. Theflow in river channels was calculated by using simple algorithm: the amount of waterthat enters the node is equal to amount that leaves node.

266 Cellular Automata - Simplicity Behind Complexity

www.intechopen.com

• Channel forming — the algorithm looks for cells with flow rate fij less than throughputrij. Such the cells are a splitting point. The route of new channel is calculated based onaltitude (initial altitude and thickness of peat layer).

Fig. 7. Sample results of simulation anastomosing river with Graph of Cellular Automataparadigm.

The Graph of Cellular Automata was applied for modelling tumour induced angiogenesiswith almost no changes in general structure Topa (2006; 2008); Topa & Dzwinel (2009).

Cellular Automata model the tissue. Some cells are initially set to represent cancer cells. Inthis model cancer development processes were not included. Cancer cells do not multiplyduring the simulation. The cancer cells are sources of various tumour angiogenic factors(e.g. VEGF, PDGF-B, Angiopoeitin) that are distributed over the mesh.

Graph of Cellular Automata is constructed over the regular mesh of cellular automata andrepresents vascular systems. Cells that belong to graph are the source of nutrients andoxygen in regular the mesh of automata. These two substances are spread in mesh tothe neighbouring cells. The cells that are nourished sufficiently stop producing tumourangiogenic factors. Graph that represents vascular systems develops in two ways:

267Network Systems Modelled by Complex Cellular Automata Paradigm

www.intechopen.com

• growth at “tip“ cells — cells in the graph are marked as ”tip” when they are locatedat the end of sprout. During simulation algorithms checks the tip cells and if necessaryconditions are fulfilled next neighbouring cells are added to the graph. The new cell arenow the “tip” cell.

• branching — when the necessary conditions are fulfilled, cell that belong to graph cancreate branch (it cannot be a "tip" cell). The algorithm searches in its neighbourhood forcell that are added to graph and marked as “tip” cells for this branch. The developmentof the sprout proceed independently from parent vessels.

When growing vessel meets the other they join creating anastomosis.

Newly created vessels have to mature what means covering by perycites and smoothmuscle cells. Only mature vessels can transport blood, supply nutrient to surroundingtissue and branch.

The model can be defined in a similar way as MANGraCA:

CAANG =< Zn, GCA, XK , S, δ > where :

• Zn — a collection of cells ordered as a square, hexadecimal (n = 2) or cubical (n = 3) mesh.,

• GCA — directed and acyclic graph defined as (VG, EG), where VG ⊂ Zn and EG ⊂ Zn × Zn

are finite sets of vertices and edges, respectively,

• XK(i, j) — neighbourhood for the (i, j) cell in the regular mesh of automata,

• S — is the set of state vectors corresponding to each cell: S = Sm × Sg,

– Sm — represents the following states corresponding to all the cells in the regular CAmesh:

* tij — state of a single tumour cell, refers to its level of supplies with nutrients,

* ta fij — TAFs concentration,

* nij — nutrient (oxygen) concentration,

* perij — perycites and smooth muscle cells concentration,

* aa fij — AAF (anti-angiogenic factors) concentration.

– Sg — represents states corresponding to the cells that belong to the Graph of CellularAutomata:

* ageij — ”age”, maturation level,

* tipij — indicate ”tip” cell (boolean),

* presij — pressure value,

* f lowij — flow value;

The model is implemented in similar manner as the model of anastomosing river. It consist oftwo modules, one for calculation on regular mesh of automata and second that process graphstructure:

Cellular Automata : some cells are marked as tumour cells and they are the sources of TAF(Tumour Angiogenic Factors). Cells that belong to graph act as sources of nutrients. Thefollowing processes are modelled in this module:

• TAF distribution,

• nutrients distribution — tumour cells that are supplied with nutrients stop producingTAF,

268 Cellular Automata - Simplicity Behind Complexity

www.intechopen.com

Fig. 8. Configuration with two primary vessels and cancer cells located in the center of area.Vessels are coloured according to flow value — blue means no flow, while red reflectmaximum flow.

• distribution of other substances that can be included to the model (angiogenicinhibitors, perycites etc.)

Graph of Cellular Automata consists of the following processes:

• updating maturation level,

• calculating flows in vessels,

• developing sprouts — for each “tip” cells in the graph their successor is calculated,

• creating new sprouts — new sprout is created in mature cells with some probability(other conditions can be also included),

Fig. 9. Hexagonal mesh can be also applied instead of rectangular. Network segments arecoloured according to flow value.

269Network Systems Modelled by Complex Cellular Automata Paradigm

www.intechopen.com

Vascular system represented by graph structure allows for convenient blood flow calculation.The model uses Poiseuille equation for calculating flow in vascular segments Mcdougall et al.(2002).

Qij =πR4

ij∆Pij

8μLij(2)

• Rij — segment diameter,

• Lij — segment length,

• μ — viscosity,

• ∆Pij = Pi − Pj — pressure difference between nodes i and j.

Fig. 10. Blood flow is calculated only in branches that created closed loops.

After upgrading graph structure the model calculates new flow distribution in network ofvessel. It consist of three steps:

1. searching for closed loops in graph — blind sprouts are not take into consideration,

2. setting and solving system of equation for new pressures distribution in graph nodes,

3. calculating flows in graph according to pressures distribution

The framework are very flexible and we can use in the model 3D mesh (see Fig. 12). Hexagonalmesh can be also applied instead of rectangular (see Fig. 9). The model can be easily extendedby new factors and processes. Figure 12 demonstrate configuration with source of angiogenicinhibitors which block vessel forming .

270 Cellular Automata - Simplicity Behind Complexity

www.intechopen.com

Fig. 11. Blood flow calculated in model of tumour induced angiogenesis. Vessels are colouredaccording to flow rate.

4. Conclusions

What is now what we called Cellular Automata? Initially Cellular Automata was treatedrather as interesting mathematical model of complexity than tool for simulating complexphysical phenomena. Their complex behaviour was the area of intensive and valuable studieson deterministic chaos and complexity. However when we try to reflect some real phenomenathat classical definition appeared to be too limited. Some modifications were introducedto better encode modelled phenomena in Cellular Automata virtual “physics“. Over time,modifications were becoming more serious. Is it still Cellular Automata? Let us enumeratewhat is in common in most models that use name ”Cellular Automata“:

• mesh, usually regular,

• evolution governed by rules,

• local neighbourhood (sometimes range is bigger then nearest cells),

It seems that those three features still are in common for models and modeling tools that useattribute ”Cellular Automata“.Sometimes Cellular Automata models become very close to Agent-Based Models (ABM)approach (sometimes referred as multi-agent systems). However, it should be noticed thatagents are characterized by much more complex and sophisticated behaviour of individualentities.It is worth to emphasize that Cellular Automata paradigm its popularity as computersimulations and modelling tool partially owes the fact that it fits very well to computerimplementation:

1. rules can be easily encoded in programming languages,

2. regular mesh of cells can be easily encoded in most typical data structures,

271Network Systems Modelled by Complex Cellular Automata Paradigm

www.intechopen.com

Fig. 12. Tumour induced angiogenesis in 3D. Cancer cells are located at the upper side of thebox. Additionally sources of angiogenic inhibitors were located in the center of the box.

3. mesh and local neighbourhood eliminates laborious necessity of determining whichelements interacts each other.

4. there is no round-off errors, numerical instabilities and so on...

In this chapter, the application of the Cellular Automata paradigm for modelling networksystems was presented. The combination of Cellular Automata and graph structure wassuccessfully applied for simulating phenomena that belong to general class of networksystems located in consuming or producing environment. The examples show how broadmeaning has Cellular Automata now. It should be also encouragement for further experimentswith this useful paradigm that cannot be constrained by the definitions.

5. Acknowledgements

The author thanks to Witold Dzwinel (Institute of Computer Science, AGH University ofScience and Technology) for his contribution to these works. The researches were partiallysupported by Institute of Computer Science, AGH University of Science and Technology(project no. 11.11.120.865) and also partially supported by the Polish Ministry of Educationand Science (project no. NN 519 579 338).

6. References

Baxter, G. & Behringer, R. (1991). Cellular automata models for the flow of granular materials,Physica D: Nonlinear Phenomena 51(1-3): 465–471.

272 Cellular Automata - Simplicity Behind Complexity

www.intechopen.com

URL: http://www.sciencedirect.com/science/article/B6TVK-46MV07J-19/2/058ea7eab54b1774410ee951bf6e69e9

Blue, V., Embrechts, M. & Adler, J. (1997). Cellular automata modeling of pedestrianmovements, Vol. 3, pp. 2320–2323 vol.3.

Cannataro, M., Gregorio, S. D., Rongo, R., Spataro, W., Spezzano, G. & Talia, D. (1995).A parallel cellular automata environment on multicomputers for computationalscience, Parallel Computing 21(5): 803–823.

Carmeliet, P. (2005). Angiogenesis in life, disease and medicine, Nature 438(7070): 932–936.10.1038/nature04478.

Chopard, B. & Droz, M. (1998). Cellular Automata Modeling of Physical Systems, Alea-SaclayMonographs and Textes in Statistical Physics, Cambridge University Press.

D’Ambrosio, D., Gregorio, S. D., Gabriele, S. & Gaudio, R. (2001). A cellular automata modelfor soil erosion by water, Physics and Chemistry of the Earth, Part B: Hydrology, Oceansand Atmosphere 26(1): 33–39.URL: http://www.sciencedirect.com/science/article/B6VPV-45KWJ8F-8/2/e17522d634e01ce1769f19d5fe1cf975

Ferrara, N. & Kerbel, R. S. (2005). Angiogenesis as a therapeutic target, Nature438(7070): 967–974. 10.1038/nature04483.

Frisch, U., Hasslacher, B. & Pomeau, Y. (1986). Lattice-gas automata for the navier-stokesequation, Phys. Rev. Lett. 56(14): 1505–1508.

Gobron, S., Altekin, A., Bonafos, H. & Thalmann, D. (2010). Gpgpu computation andvisualization of three-dimensional cellular automata, The Visual Computer 27: 67–81.

Gradzinski, R., Baryla, J., Doktor, M., Gmur, D., Gradzinski, M., Kedzior, A., Paszkowski, M.,Soja, R., Zielinski, T. & Zurek, S. (2003). Vegetation-controlled modern anastomosingsystem of the upper narew river (ne poland) and its sediments, Sedimentary Geology157(3-4): 253–276.

Graner, F. & Glazier, J. A. (1992). Simulation of biological cell sorting using a two-dimensionalextended Potts model, Physical Review Letters 69: 2013–2016.

Hardy, J., de Pazzis, O. & Pomeau, Y. (1976). Molecular dynamics of a classical lattice gas:Transport properties and time correlation functions, Phys. Rev. A 13(5): 1949–1961.

Kier, L. B. (2000). A cellular automata model of bond interactions among molecules, Journal ofChemical Information and Computer Sciences 40(5): 1285–1288.

Kier, L. B., Cheng, C.-K., Tute, M. & Seybold, P. G. (1998). A cellular automata model of aciddissociation, Journal of Chemical Information and Computer Sciences 38(2): 271–275.

Margolus, N., Toffoli, T. & Vichniac, G. (1986). Cellular-automata supercomputers forfluid-dynamics modeling, Phys. Rev. Lett. 56(16): 1694–1696.

Markus, M., Bohm, D. & Schmick, M. (1999). Simulation of vessel morphogenesis usingcellular automata, Mathematical Biosciences 156(1-2): 191–206.URL: http://www.sciencedirect.com/science/article/B6VHX-3W374Y2-9/2/d03620d18d402fe799635911fa3dab5c

Masselot, A. & Chopard, B. (1996). Cellular automata modeling of snow transport bywind, in J. Dongarra, K. Madsen & J. Wasniewski (eds), Applied Parallel ComputingComputations in Physics, Chemistry and Engineering Science, Vol. 1041 of Lecture Notesin Computer Science, Springer Berlin / Heidelberg, pp. 429–435.

Mcdougall, S. R., Anderson, A. R. A., Chaplain, M. A. J. & Sherratt, J. A. (2002).Mathematical modelling of flow through vascular networks: Implications fortumour-induced angiogenesis and chemotherapy strategies, Bulletin of MathematicalBiology 64(42): 673–702.

273Network Systems Modelled by Complex Cellular Automata Paradigm

www.intechopen.com

Neumann, J. V. (1966). Theory of Self-Reproducing Automata, University of Illinois Press,Champaign, IL, USA.

Reis, E., Santos, L. & Pinho, S. (2009). A cellular automata model for avascular solid tumorgrowth under the effect of therapy, Physica A: Statistical Mechanics and its Applications388(7): 1303–1314.URL: http://www.sciencedirect.com/science/article/B6TVG-4V2NP1G-4/2/d7ec3cee53804a2bfa771c79ba7875bf

Somers, J. & Rem, P. (1989). A parallel cellular automata implementation on a transputernetwork for the simulation of small scale fluid flow experiments, in G. van Zee &J. van de Vorst (eds), Parallel Computing 1988, Vol. 384 of Lecture Notes in ComputerScience, Springer Berlin / Heidelberg, pp. 116–126.

Steve, N. M., Webb, S. & Othmer, H. G. (2004). Mathematical modeling of tumor-inducedangiogenesis, J. Math. Biol 49: 111–187.

Toffoli, T. & Margolus, N. (1987). Cellular automata machines: a new environment for modeling,MIT Press, Cambridge, MA, USA.

Tonini, T., Rossi, F. & Claudio, P. P. (n.d.). Molecular basis of angiogenesis and cancer, Oncogene22(42): 6549–6556.

Topa, P. (2006). Towards a two-scale cellular automata model of tumour-induced angiogenesis,in S. E. Yacoubi, B. Chopard & S. Bandini (eds), ACRI, Vol. 4173 of Lecture Notes inComputer Science, Springer, pp. 337–346.

Topa, P. (2008). Dynamically reorganising vascular networks modelled using cellularautomata approach, Cellular Automata, 8th International Conference on CellularAutomata for Reseach and Industry, ACRI 2008, Yokohama, Japan, September 23-26, 2008.Proceedings, Vol. 5191 of Lecture Notes in Computer Science, Springer, pp. 494–499.

Topa, P. & Dzwinel, W. (2003). Consuming environment with transportation networkmodelled using graph of cellular automata, in R. Wyrzykowski, J. Dongarra,M. Paprzycki & J. Wasniewski (eds), PPAM, Vol. 3019 of Lecture Notes in ComputerScience, Springer, pp. 513–520.

Topa, P. & Dzwinel, W. (2009). Using network descriptors for comparison of vascularsystems created by tumour-induced angiogenesis, Theoretical and Applied Informatics21(2): 83–94.

Topa, P., Dzwinel, W. & Yuen, D. A. (2006). a Multiscale Cellular Automata Model forSimulating Complex Transportation Systems, International Journal of Modern PhysicsC 17: 1437–1459.

Topa, P. & Paszkowski, M. (2001). Anastomosing transportation networks, in R. Wyrzykowski,J. Dongarra, M. Paprzycki & J. Wasniewski (eds), PPAM, Vol. 2328 of Lecture Notes inComputer Science, Springer, pp. 904–912.

Wang, H., Nie, G. & Fu, K. (2008). Cellular automata simulation of the growth of bonetissue, ICNC ’08: Proceedings of the 2008 Fourth International Conference on NaturalComputation, IEEE Computer Society, Washington, DC, USA, pp. 421–424.

Was, J. (2005). Cellular automata model of pedestrian dynamics for normal and evacuationconditions, ISDA ’05: Proceedings of the 5th International Conference on IntelligentSystems Design and Applications, IEEE Computer Society, Washington, DC, USA,pp. 154–159.

Wolfram, S. (2002). A New Kind of Science, Wolfram Media.URL: http://www.amazon.com/exec/obidos/ASIN/1579550088/ref=nosim/rds-20

Wooldridge, M. (2009 ). An Introduction to Multiagent Systems, Wiley.

274 Cellular Automata - Simplicity Behind Complexity

www.intechopen.com

Cellular Automata - Simplicity Behind ComplexityEdited by Dr. Alejandro Salcido

ISBN 978-953-307-230-2Hard cover, 566 pagesPublisher InTechPublished online 11, April, 2011Published in print edition April, 2011

InTech EuropeUniversity Campus STeP Ri Slavka Krautzeka 83/A 51000 Rijeka, Croatia Phone: +385 (51) 770 447 Fax: +385 (51) 686 166www.intechopen.com

InTech ChinaUnit 405, Office Block, Hotel Equatorial Shanghai No.65, Yan An Road (West), Shanghai, 200040, China

Phone: +86-21-62489820 Fax: +86-21-62489821

Cellular automata make up a class of completely discrete dynamical systems, which have became a coresubject in the sciences of complexity due to their conceptual simplicity, easiness of implementation forcomputer simulation, and their ability to exhibit a wide variety of amazingly complex behavior. The feature ofsimplicity behind complexity of cellular automata has attracted the researchers' attention from a wide range ofdivergent fields of study of science, which extend from the exact disciplines of mathematical physics up to thesocial ones, and beyond. Numerous complex systems containing many discrete elements with localinteractions have been and are being conveniently modelled as cellular automata. In this book, the versatilityof cellular automata as models for a wide diversity of complex systems is underlined through the study of anumber of outstanding problems using these innovative techniques for modelling and simulation.

How to referenceIn order to correctly reference this scholarly work, feel free to copy and paste the following:

Pawel Topa (2011). Network Systems Modelled by Complex Cellular Automata Paradigm, Cellular Automata -Simplicity Behind Complexity, Dr. Alejandro Salcido (Ed.), ISBN: 978-953-307-230-2, InTech, Available from:http://www.intechopen.com/books/cellular-automata-simplicity-behind-complexity/network-systems-modelled-by-complex-cellular-automata-paradigm

© 2011 The Author(s). Licensee IntechOpen. This chapter is distributedunder the terms of the Creative Commons Attribution-NonCommercial-ShareAlike-3.0 License, which permits use, distribution and reproduction fornon-commercial purposes, provided the original is properly cited andderivative works building on this content are distributed under the samelicense.